Killing forms are bilinear forms most well-known in the context of Lie algebras. Their definition has been extended to braided-Killing forms on braided-Lie algebras, and applying this to certain braided-Lie algebras based on finite groups yields an expression in terms of centralizers in the group.
In this talk, based on a collaboration with Kevin Piterman, we discuss some problems concerning the non-degeneracy and irreducibility of these forms on finite groups. We cover classes of involutions in simple groups of Lie type and Lie rank 1, and study the case of unipotent elements in PSL(2, q) and PSU(3, q).